An important variant of this problem is the analogous question in bipartite graphs with VC-dimension at most \(d\) , where \(d\)  is a fixed integer not larger than \(k\) . The VC-dimension of a set system \(\mathcal {F}\)  on the ground set \(V\)  is the largest integer \(d\)  for which there exists a \(d\) -element set \(S \subset V\)  such that for every subset \(B \subset S\) , one can find a member \(A \in \mathcal {F}\)  with \(A \cap S = B\) . The VC-dimension, introduced by Vapnik and Chervonenkis [1]}, is one of the most useful combinatorial parameters that measures the complexity of graphs and hypergraphs. Over the years it has proven tremendously important in many areas in (and out of) combinatorics and mathematics, in general. We refer to [2]} for a nice discussion (and a further remarkable application). In order to define the VC-dimension of a bipartite graph \(G = (A,B,E)\)  with vertex set \(A \cup B\)  and edge set \(E \subset A \times B\) , the standard convention is to make a choice between \(A\)  and \(B\) , and then define the VC-dimension of \(G = (A,B,E)\)  with respect to \(A\) to be the VC-dimension of the set system of neighborhoods of vertices \(b \in B\)  (regarded as subsets of \(A\) ), and vice versa (see also [3]}). In this paper, we will always choose the first side of the bipartition \(A\)  as the ground set, and so we shall say that the VC-dimension of a bipartite \(G = (A,B,E)\) is the VC-dimension of \(G\)  with respect to \(A\) .
